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Fault Diagnosis and Prognosis of a Brushless DC motor using a
Model-based Approach
Joaquim Blesa, Joseba Quevedo, Vicenç Puig, Fatiha Nejjari, Raúl Zaragoza and Alejandro Rolán
Supervision, Safety and Automatic Control Research Center (CS2AC)
Universitat Politècnica de Catalunya (UPC),Campus de Terrassa,
Gaia Building, Rambla Sant Nebridi, 22
08222 Terrassa, Barcelona
{joaquim.blesa, joseba.quevedo, vicenc.puig, fatiha.nejjari, alejandro.rolan}@upc.edu
ABSTRACT
This paper proposes a model-based fault diagnosis and
prognosis approach applied to brushless DC motors
(BLDC). The objective is an early detection of mechanical
and electrical faults in BLDC motors operating under a
variety of operating conditions. The proposed model-based
method is based on the evaluation of a set of residuals that
are computed taking into account analytical redundancy
relations. Fault diagnosis consist of two steps: First,
checking if at least one of the residuals is inconsistent with
the normal operation of the system. And, second, evaluating
the set of the residuals that are inconsistent to determine
which fault is present in the system. Fault prognosis consists
of the same two steps but instead of considering current
inconsistencies evaluates drift deviations from nominal
operation to predict futures residual inconsistencies and
therefore predict future fault detections and diagnosis. A
description of various kinds of mechanical and electrical
faults that can occur in a BLDC motor is presented. The
performance of the proposed method is illustrated through
simulation experiments.
1. INTRODUCTION
In the last years, a growing industrial interest in fault
diagnosis and more recently in fault prognosis is perceived
in the research community. In the scientific literature, fault
diagnosis is nowadays a quite mature field with well-
established approaches (see as e.g. Blanke el al., 2006,
Ding, 2008, Escobet et al., 2019). However, prognosis is
still in a less mature stage where most of the approaches
existing are quite ad-hoc to the application.
In this paper, the extension of the well-established model-
based fault diagnosis approaches to deal with prognosis will
be proposed. This idea has already been explored by several
researchers (Roychoudhury et al., 2006; Yu et al., 2011). In
fact, fault diagnosis methods for incipient faults are suitable
to be extended for prognosis.
This paper proposes a model-based fault diagnosis and
prognosis approach applied to brushless DC motors. The
objective is an early detection of rotor and load faults in
BLDC motors operating under a variety of operating
conditions. This involves recognizing the rotor and fault
signatures produced in BLDC motors, and estimating the
severity of the fault. A description of various kinds of rotor
and load faults that can occur in a BLDC motor is presented.
The effects of various types of faults on the motor current or
voltage, which represent an alternative to vibration-based
diagnostics, are investigated and illustrated through
simulation experiments.
The structure of the paper is the following: Section 2
presents the problem statement and an overview of the
proposed solution. Section 3 presents the fault diagnosis and
prognosis approach. Section 4 describes the case study
based on a brushless DC motor including the model and the
fault scenarios considered. Section 5 presents the results.
Finally, Section 6 draws the main conclusions.
2. PROBLEM STATEMENT: MODEL-BASED
DIAGNOSIS/PROGNOSIS
2.1. Problem statement
Given a model of the system and a set of output observed
variables, ky , and the set of inputs, ku , consistency tests
can be derived from an analytical redundancy relation Joaquim Blesa et al. This is an open-access article distributed under the
terms of the Creative Commons Attribution 3.0 United States License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original author and source are credited.
EUROPEAN CONFERENCE OF THE PROGNOSTICS AND HEALTH MANAGEMENT SOCIETY 2020
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(ARR) by generating a computational residual in the
following way (see Blanke et al., 2006):
, 0,i i k kr y u
where i is called the residual ARR expression. The set of
ARRs can be represented as
, 0, 1, , ,i i k k rr y u i n (2)
where rn is the number of ARRs obtained after applying the
structural analysis.
Let be the set of faults that must be monitored.
In this paper, a model based approach using residuals
obtained from the ARRs (2) that will be used for dealing
with the fault diagnosis and prognosis problems is
introduced in the following.
Problem 1. “Fault diagnosis”. Fault diagnosis aims at
detecting and identifying the fault that is affecting a system
in a fault scenario given a sequence of values of the output
observed variables, ky , and the set of inputs, ku , and the
considered set of faults and ARRs in (2).
That is, fault diagnosis is about detecting and isolating a
fault that has already occurred in the system. On the other
hand, fault prognosis is about anticipating the appearance of
a fault in time. Thus, the second problem addressed in this
paper is defined as follows.
Problem 2. “Fault prognosis”. Fault prognosis aims at
predicting the appearance of fault in the system in a given
time horizon from the current time given a sequence of
values of the output observed variables, ky , and the set of
inputs, ku , and the considered set of faults and ARRs in
(2).
2.2. Overview of the proposed approach
The proposed approach is based on the following modules:
- residual generation via the ARRs (2). Using the
sequence of values of the output observed
variables, ky , and the set of inputs, ku , the ARRs
will be evaluated using their mathematical
expressions generating a set of residuals.
- residual analysis via two processes: every ARR
will be evaluated against its corresponding
detection threshold. If the threshold is violated, a
fault is indicated. A further analysis is required
matching the violated residuals against the fault
signature matrix to discover the fault affecting the
system among the set of considered faults . If
the residual is not violated, the residual value in a
time horizon is evaluated to detect some trend. If a
significant trend is detected, it is projected towards
the future to determine the time required for
residual to violate its threshold. From this
information, the fault prognosis algorithm will be
able to predict the appearance of a fault in a given
time horizon.
3. PROPOSED SOLUTION: FAULT DIAGNOSIS/PROGNOSIS
USING ANALYTICAL REDUNDANCY RELATIONS
Using the set of computable ARR residuals (1), the fault
detection module must check at each time instant whether or
not the ARRs are consistent with the observations. Under
ideal conditions, residuals are zero in the absence of faults
and non-zero when a fault is present. However, modeling
errors, disturbances and noise, in complex engineering
systems are inevitable, and hence it appears the necessity of
applying robust fault detection algorithms.
3.1. Residual generation
Taking into account bounded uncertainties, the residuals (1)
can be rewritten as follows:
, ,i i k k kr y u (3)
with: k , where is the interval box
n , that includes all uncertainties (i.e.
disturbances/model errors and measurement noise). Then,
fault detection can be formulated as checking the
consistency of (3) using a set-membership approach (Puig et
al., 2013).
Definition 1. “Consistency checking”. Given the
residuals described by (3) and a sequence of measured
inputs ku and outputs ky of the real system at time k, they
are consistent with measurements and uncertainty bounds if
there exist a set of sequences k that satisfies ir 0 .
Thus, according to this definition, a residual consistency
is equivalent to check if 0 ( ) ir k where ( )ir k is the
interval that bounds the effect of uncertainty in the residual
(3).
3.2. Residual analysis
Definition 2. Fault detection. Given a sequence of
observed inputs ku and outputs ky of the real system, a
fault is said to be detected at time k if there does not exist a
set of uncertainty sequences k to which the set of
ARRs is consistent.
In case that 0 ( ) ir k , although a fault is detected, the
evolution of the residual can be forecasted h-steps ahead using
the Brown’s Double Exponential Smoothing is used (S. Hansun,
2016) that is based on the following multi-step forecast formula
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( ) ( ) (1 ) ( 1) i i ir t r t r t
( ) ( ) (1 ) ( 1) i i ir t r t r t (4)
ˆ ( | ) 2 ( ) 1 ( )1 1
i i i
h hr k h k r k r k
where h is the forecasting horizon and α the smoothing
parameter. The parameter α is obtained from historical data
using parameter estimation by means of the least squares
algorithm using a moving horizon time window L.
The Remaining Useful Life (RUL) forecasted by the
residual ir based on the corresponding threshold given by
the residual interval ( )ir k can be determined as follows
0ˆ| 0 ( | ) i i iRUL r k RUL k (5)
where ˆ( | )r k RUL k is the RUL-step ahead forecast at
time k of the corresponding predictive model (4).
3.3. Fault diagnosis
According to Definition 2, a fault is detected at time k when
0 ( ) ir k . The information provided by the consistency
checking is stored as fault signal ki :
0 0 ( )
1 0 ( )
ii
i
if r kk
if r k . (6)
The fault isolation module used in this paper derives from
the one proposed in (Puig et al., 2013). The first component
is a memory that stores information about the fault signal
occurrence history and the fault detection module updates it
cyclically (Figure 1).
Figure 1. Fault isolation module components
The pattern comparison component compares the memory
contents with the stored fault patterns. The standard Boolean
fault signature matrix concept (Escobet et al., 2019) is
generalized taking into account more fault signal properties
(signs, fault sensitivities, etc). The last component
represents the decision logic part of the method which aim
is to propose the most probable fault candidate.
3.4. Fault prognosis
The detection of fault is forecasted by evaluating the
minimum RUL of the set of residual RULs. The RULi of
each residual is determined as the time for the current k such
that ˆ0 ( ) i ir k RUL according to (5). Every residual has a
forecasted fault signal associated at its corresponding RUL:
ˆ 1 i ik RUL .
The fault prognosis module uses the same logic that the
fault isolation algorithms but using the forecasted signals
ˆ i ik RUL . The forecasted isolation fault at time k can
be obtained by analyzing the activated residuals at the end
of the corresponding RULs (5).
4. CASE STUDY DESCRIPTION
4.1. BLDC motor model
The constitution of a BLDC motor bears a resemblance to a
permanent magnet synchronous machine (PMSM): surface-
mounted permanent magnets in the rotor and a 3-phase wye
connected winding in the stator. However, its back electro-
motive force (emf) induced in the stator is not sinusoidal, as
in PMSMs, but trapezoidal (Pillay et al., 1989).
Figure 2(a) (adapted from Xia, 2002) shows the 3-phase
equivalent circuit of a BLDC motor powered by a full-
bridge inverter, which transforms the DC voltage, Vdc, into
3-phase AC voltages, vabc, by means of the switching pattern
of 6 Insulated Gate Bipolar Transistors (IGBTs) (S1,…,S6)
according to the signals given by 3 Hall sensors that detect
the rotor position. The following assumptions are made: (a)
the stator windings are symmetrical (i.e., they have the same
number of turns with same inductance and resistance) and
equally distributed in the stator; (b) there is no saturation in
the magnetic core; (c) Eddy currents and hysteresis losses
are negligible, so there are no iron losses; (d) the armature
reaction is ignored; (e) the air-gap magnetic field has a
trapezoidal shape, which creates trapezoidal back emfs in
the stator windings (as seen in Figure 2(b)); (f) no damper
windings are needed in the rotor because permanent
magnets have high resistivity; and (g) the permanent
magnets are surface-mounted, so there is no saliency, i.e.,
the airgap thickness is constant; and (h) there is no cogging
torque (note that in a salient-pole configuration, slots and
teeth are passed by the field magnet and when the rotor
moves cogging is generated as a result of reluctance
new event
occurred?
update
memory
residuals
evaluation
evaluation
factor01
evaluation
factorsign
evaluation
factorsensit
evaluation
factortime
evaluation
factororder
decision
logic
Memory
Component
Pattern
Comparison
Component
Decision Logic
Component
diagnosed
fault
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variation. However, in cylindrical rotors (surface-mounted
permanent magnets), there is no reluctance variation (no
teeth), so rotation is coggingless (Kenjo et al., 1985)).
According to the previous assumptions, the following
differential equations (Pillay et al., 1989) describe the
dynamic behavior of a BLDC motor with p pole pairs (note
that the electrical equations correspond to the equivalent
circuit shown in Figure 2(a) and for the mechanical equation
the 1-mass model for the shaft has been considered):
Figure 2. BLDC motor (adapted from Xia, 2002). (a) 3-phase
equivalent circuit (motor-sign convention) powered by a full-
bridge inverter, (b) trapezoidal per-phase back emfs and per-phase
currents in the stator windings, and (c) equivalent circuit under the
2-phase conduction mode (each π/3 rad). Annotation:
C = capacitance, eabc = per-phase back emf induced in the stator,
i = current of the simplified circuit, iabc = per-phase stator current,
ke = back emf coefficient, L = per-phase self-inductance,
Leq = equivalent inductance of the simplified circuit, M = mutual
inductance, n = neutral point of the star connection, p = pole pairs,
R = per-phase resistance, Req = equivalent resistance of the
simplified circuit, S1…S6 = IGBT-based controlled switches,
vabc = per-phase stator voltage, Vdc = DC voltage, θr = rotor angle,
λ'm = amplitude of the flux linkage by the permanent magnets,
ωr = rotor speed.
a a a a
b b b b
c c c c
re L r r e a a c c c r
rr
0 0d
;
0 0 d
0 0
dωω ω
d
dθω
d
b
v R i L M M i e
v R i M L M i et
v R i M M L i e
T T J B T e i e i e it
t
(7)
where vabc is the per-phase stator voltage, iabc is the per-
phase stator current, R is the per-phase resistance, L is the
per-phase self-inductance, M is the mutual inductance
between each pair of windings, Te is the electromagnetic
torque, TL is the resisting (or load) torque, J is the moment
of inertia of the rotational system (BLDC motor and load),
Br is the damping (or viscous friction) coefficient, ωr is the
rotor speed, θr is the rotor angle (whose value is detected by
the Hall sensors to obtain the rotor position), d/dt is the
differential operator and eabc is the per-phase back emf
induced in the stator windings caused by the rotor
permanent magnets, whose waveform is shown in
Figure 2(b), and whose equation is:
' 2 2( ) ( ) ( )
3 3
TT
a b c r m a r a r a re e e p f p f p f p
(8)
where p is the number of pole pairs, is the amplitude of
the flux linkage established by the permanent magnets in the
rotor as viewed from the stator windings, and fa (pθr) is the
trapezoidal function (defined between 1 and –1) shown in
Figure 2(b) for phase a. Note that the product
corresponds to the electrical angle, θe, of the BLDC motor.
It should be noted from the currents iabc shown in
Figure 2(b) that there are only 2-phase windings conducting
in each interval of π/3 rad (or 60 deg.), which is known as
the 2-phase conduction mode of a BLDC motor (Xia, 2002).
Let consider that the interval π/3 marked in Figure 2(b),
where phases a and b are conducting and phase c is not
(switches S1 and S6 in Figure 2(a) are turned on and the rest
ones are turned off). Under these circumstances, the
electrical behavior of the BLDC motor is given by the
following equation:
2 2( ) 2ab dc a
dc eq eq e r
div v Ri L M e
dt
div R i L k
dt
(9)
where i = ia = –ib, ea = –eb, Vdc is the DC bus voltage,
Req = 2R is the equivalent line resistance, Leq = 2(L – M) is
the equivalent line inductance, and is the
coefficient of the back emf. The equivalent circuit that
corresponds to (9) is shown in Figure 2(c), which is similar
to the equivalent circuit of a series-excited dc motor. The
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electromagnetic torque equation given in (7) can now be
expressed as: '2e m TT p i k i (10)
where
is the torque coefficient.
Finally, the dynamic behavior of a BLDC motor under the
2-phase conduction mode is given by the electrical equation
(9) and by the mechanical equations (the last 2 equations in
(7), where the electromagnetic torque is simplified as (10)),
giving the following matrix system written in the state-space
equation form (dx/dt = Ax + Bu, where the state-space
variables x are the current, i, and the rotor speed, ωr):
eq e
dceqeq eq
r r LT r
10
d
ω ωd 10
R k
i i VLL L
Tt k B
JJ J
(11)
4.2. BLDC motor faults
Brushless DC motors are very reliable devices. However,
they can suffer from some failure due to overheating and
mechanical wear (Da et al, 2011) and demagnetization
(Moosavi et al., 2015). The main faults can generally be
categorized in stator faults, rotors faults, inverter faults and
bearing faults (Rajagopalan et al., 2006). They can also be
classified into electrical faults and mechanical faults.
Electrical faults typically involve short-circuited faults of
the stator windings and resistive unbalance faults. This kind
of fault may produce an unbalance of stator voltages and
currents, an increased torque pulsation, a decreased average
torque or increased losses and excessive heating.
On the other hand, mechanical faults include eccentricity
and bearing faults. They are due to a degradation process in
the bearings, destruction of the bearing cage, balls and
rollers, and rotor shaft deformation, among others. This kind
of fault may result in increased friction for the brushless dc
motor. Moreover, necessary sensors for position or current
measurement, can fail as well.
In this paper, the following set of faults will be considered:
- faults in sensors: position (fθ) and current (fi)
- parametric faults: resistance (fR), inductance (fL),
friction (fB) and inertia (fJ)
- systems faults: voltage supply (fVdc), load (fLoad)
5. RESULTS
5.1. BLDC motor residuals
Considering the state space model (11), and using the
method of generation of structured residuals proposed in
(Isermann, 2006), the following residuals can be generated:
1 eq eq e r dc
2 r r T L
23 eq eq eq eq e T dc e L
24 r eq eq eq eq e T dc T eq eq L
( ) ω
( ) ω
( ) ( )
( ) ω
r s R I s L sI s k s V s
r s s Js B k I s T s
r s I s JL s R J BL s BR k k V s Js B k T s
r s s JL s R J BL s BR k k
V s k L s R T s
(12)
These residuals are discretized in time for real-time
implementation. Threshold values associated to residuals are
computed with maximum absolute values of computed
residuals in fault free scenarios.
From the previous four residuals, considering the faults
described in Section 4.2, the fault signature matrix (FSM)
presented in Table 1 can be obtained.
Table 1. Fault signature matrix
fR fL fVdc fB fJ fθ fi fLoad
r1 1 1 1 0 0 1 1 0
r2 0 0 0 1 1 1 1 1
r3 1 1 1 1 1 0 1 1
r4 1 1 1 1 1 1 0 1
5.2. Results of diagnosis
Scenario 1: “Encoder stuck fault”
The position encoder stuck fault scenario consists in losing
20.000 pulses of the encoder (with a resolution of 1024
pulses per revolution) during a small time interval, in this
case from 10 to 11 seconds (see Figure 3). As can be seen in
the fault signature matrix, the fault affects the residuals r1, r2
and r4 but not r3. Notice that there is no delay in detecting
the fault in residuals r1, r2 and r4 (see Figs 4, 5 and 7).
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Figure 3. Position sensor fault scenario
Figure 4. Residual 1 evolution
Figure 5. Residual 2 evolution
Figure 6. Residual 3 evolution
Figure 7. Residual 4 evolution
Analysing the sign of the affected residuals (factor sign in
Figure 1), it can be observed that an encoder stuck situation
produces a negative activation of r1, r2 and r4. It can also be
seen that these residuals are very sensitive to this fault,
because in only one second of encoder pulses loss the motor
produces an increase of 4000% of the value of the residual
with respect to the threshold (factor sensitivity in Figure 1).
However, this is maintained only during the loss of pulses
(stuck situation). Thus, a mechanism of activation holding
must be implemented (memory component in Figure 1),
when the residual overcomes the threshold, to inform about
the occurrence of the fault.
5.3. Results of prognosis
Scenario 2: “Incipient leak in the current sensor”
An incipient fault in the current sensor produces a big error
until 0.9A after 100 seconds of the simulation scenario. This
fault starts at 10 seconds and affects linearly the magnitude
of the residuals r1, r2 and r3 but does not affect residual r4
(see Figs 9, 11, 13 and 14).
All the residuals show a notable level of noise but
considering only the mean values of the residuals a delay of
approximately 12 seconds can be observed for the activation
of the residuals r1 and r3 (Figs. 10 and 14) and 15 seconds
for r2 (see Figure 12). In the analysis of the sign of the
affected residuals, a positive increase of the big error in the
current sensor produces a positive activation of r1 and r3 (see
Figure 9 and Figure13) and a negative one of r2 (see Figure
11). Concerning the sensitivity analysis, one can see that
the residuals increase in the same manner than the fault (0.9
unities at the end of the scenario) and overcome with a
700% the threshold at the end of the simulated scenario.
This fault produces a constant degradation of the residuals
which enables the use of a prognosis method to anticipate
the activation time of the residuals, that is the RUL
determiner according to (5). The RUL of each residual
allows predicting the time to reach its threshold. In this
work, the Brown double aliasing technique (4) is very
appropriate because firstly the double aliasing reduces
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significantly the noise of the residuals and secondly this
model allows estimating the Remaining Useful Life (RUL)
(5). Figs. 10, 12, 14 and 16 show that the evolution of RUL
over time is approaching the real RUL value. These figures
show many estimated RUL<0 that become false positive
residual activations, which can be easily solved eliminating
the estimated RUL below zero. For this scenario,
considering an interval of 2% (due to measurement noise),
the proposed prognosis method allows to anticipate 9
seconds the activation of r1, 4 seconds of r2 and around 3.5
seconds of r3. Consequently, using this prognosis allows
anticipating the time for the detection and isolation of this
fault as shown in Table 2.
Table 2. Prognosis results
Residual
Prognosis
r1 r2 r3 r4
RUL 9 s 4 s 3.5 s Not
affected
Fault
detection
Anticipation time of 9 seconds to the real fault
detection
Fault
isolation
Anticipation time of 3.5 seconds to the real fault
isolation
Figure 8. Current sensor fault
Figure 9. Evolution of r1 current fault
Figure 10. RUL of r1
Figure 11. Evolution of r2 current fault
Figure 12. RUL of r2
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Figure 13. Evolution of r3 current fault
Figure 14. RUL of r3
Figure 15. Evolution of r4 current fault
Figure 16. RUL of r4
6. CONCLUSIONS
This paper has proposed a model-based fault diagnosis and
prognosis approach for brushless DC motors (BLDC). A
simplified model extracted from the BLDC motor equations
has been used to generate four structural residuals. The on-
line evaluation of the inconsistency of computed residuals
allows the fault diagnosis (detection and isolation) of the
considered mechanical and electrical faults. Inconsistency of
a residual is determined if its absolute value is above a
threshold computed from non-faulty data. A Double
Exponential Smoothing filter is applied to residuals in order to
predict future inconsistencies of residuals a therefore future
faults. The proposed model-based is an alternative to
vibration-based diagnostics because residuals are computed
from analytical redundancy relations between measured and
estimated variables that are already available in BLDC
motors for control purposes. A realistic simulator has been
used to evaluate the performance of the proposed method
obtaining satisfactory results in both fault diagnosis and
prognosis for the different fault scenarios.
ACKNOWLEDGEMENT
This work has been funded by the Spanish State Research
Agency (AEI) and the European Regional Development
Fund (ERFD) through the project SCAV (ref. MINECO
DPI2017-88403-R), by the DGR of Generalitat de
Catalunya (SAC group ref. 2017/SGR/482) and by SMART
Project (ref. num. EFA153/16 Interreg Cooperation Program
POCTEFA 2014-2020). Joaquim Blesa and Alejandro
Rolán acknowledge the support from the Serra Húnter
program.
REFERENCES
Blanke, M., Kinnaert, M., Lunze, J., Staroswiecki, M.
(2006) Diagnosis and Fault-Tolerant Control. Springer,
2nd edition.
Da, Y., Shi, X., Krishnamurthy, M. (2011) Health
monitoring, fault diagnosis and failure prognosis
techniques for brushless permanent magnet machines.
In 2011 IEEE Vehicle Power and Propulsion
Conference, Chicago, IL, pp. 1–7.
Ding, S. X. (2008) Model-based Fault Diagnosis
Techniques. Springer.
Escobet, T., Bregon, A., Pulido, B., Puig, V. (2019) Fault
Diagnosis of Dynamic Systems. Springer.
Isermann, R. (2006) Fault-diagnosis systems: an
introduction from fault detection to fault tolerance.
Springer.
Kenjo, T., Nagamori, S. (1985) Permanent-magnet and
brushless DC motors. Oxford University Press.
Moosavi, S.S., A. Djerdir, Y. Ait. Amirat, D.A. Khaburi,
(2015) Demagnetization fault diagnosis in permanent
magnet synchronous motors: A review of the state-of-
EUROPEAN CONFERENCE OF THE PROGNOSTICS AND HEALTH MANAGEMENT SOCIETY 2020
9
the-art. Journal of Magnetism and Magnetic Materials,
391, pp.203–212.
Pillay, P., Krishnan, R. (1989) “Modeling, simulation, and
analysis of permanent-magnet motor drives. Part II: the
brushless dc motor drive,” IEEE Trans. Ind. App., vol.
25, no. 2, pp. 274-279.
Puig, V., Blesa, J. (2013) “Limnimeter and rain gauge FDI
in sewer networks using an interval parity equations
based detection approach and an enhanced isolation
scheme”, Control Engineering Practice, Volume 21,
Issue 2, Pages 146-170.
Rajagopalan, S., Aller, J. M., Restrepo, J. A., Habetler, T.
G., Harley, R. G. (2006) “Detection of Rotor Faults in
Brushless DC Motors Operating Under Nonstationary
Conditions”. IEEE Trans. Ind. App., vol. 42, no. 6, pp.
1464-1477.
Roychoudhury, I., Biswas, G., Koutsoukos, X. (2006). “A
Bayesian Approach to Efficient Diagnosis of Incipient
Faults,” in Prod. 17th international Workshop on
Principles of Diagnosis, Peñaranda de Duero, Spain.
S. Hansun, A new approach of Brown's double exponential
smoothing method in time series analysis, Balkan
Journal of Electrical and Computer Engineering 4
(2016) pp.75-78.
Xia, C. L. (2002). Permanent Magnet Brushless DC Motor
Drives and Controls. Wiley.
Yu, M., Wang, D., Luo, M., Huang, L. (2011). “Prognosis
of Hybrid Systems with Multiple Incipient Faults:
Augmented Global Analytical Redundancy Relations
Approach,” IEEE Trans. Syst., Man, and Cybern - part
A: Syst. and Hum., vol. 42, no. 3, pp. 1083-4427.